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Find all points of discontinuity of \(f\), where \(f\) is defined by $ f(x) = \left\{ \begin{array} {1 1} |\;x\;| + 3,& \quad\text{ if $ x $ \(\leq -3\)}\\ -2x ,& \quad \text{if $-3$ <x< 3}\\ 6x+2, & \quad\text{if $x$ \(\geq 3\)}\\ \end{array} \right. $

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Toolbox:
  • If $f$ is a real function on a subset of the real numbers and $c$ be a point in the domain of $f$, then $f$ is continuous at $c$ if $\lim\limits_{\large x\to c} f(x) = f(c)$.
Step 1:
At $x=-3$
LHL=$\lim\limits_{\large x\to -3}(\mid x\mid+3)$
$\quad\;\;=\lim\limits_{\large x\to -3}(-x+3)$
$\quad\;\;=-(-3)+3$
$\quad\;\;=3+3=6$
$f(-3)=|-3|+3=6$
RHL=$\lim\limits_{\large x\to -3^+}f(x)=\lim\limits_{\large x\to -3^+}(-2x)$
$\quad\;\;=(-2(-3))=6$
LHL=RHL=$f(-3)$
$f$ is continuous at $x=-3$
Step 2:
At $x=3$
LHL=$\lim\limits_{\large x\to 3^-}f(x)=\lim\limits_{\large x\to 3^-}(-2x)$
$\qquad\qquad\qquad\;=-2\times -3=6$
RHL=$\lim\limits_{\large x\to 3^+}f(x)=\lim\limits_{\large x\to 3^+}(6x+2)$
$\qquad\qquad\qquad\;=(6(3)+2)$
$\qquad\qquad\qquad\;=18+2$
$\qquad\qquad\qquad\;=20.$
$f(3)$ is not defined.
LHL $\neq$ RHL $\neq$ f(3)
$f$ is discontinuous at $x=3$
Step 3:
At $x=c<-3$
$\lim\limits_{\large x\to -c}(\mid x\mid+3)=-c+3=f(c)$
$\lim\limits_{\large x\to -c}f(x)=f(c)$
$\Rightarrow f$ is continuous at $x=c<-3$
Step 4:
At $x=c$ when $-3< x < 3$
$\lim\limits_{\large x\to c}(-2x)=-2c$
$\qquad\quad\quad\;=f(c)$
$\Rightarrow \lim\limits_{\large x\to c}f(x)=f(c)$
$f$ is continuous at $x=c$ where $-3< c < 3$
Step 5:
At $x=c > 3\lim\limits_{\large x\to c}(6x+2)=6c+2=f(c)$
$\Rightarrow \lim\limits_{\large x\to c}f(x)=f(c)$
$f$ is continuous at $x=c>3$
answered May 27, 2013 by sreemathi.v
 

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